On a theorem of Gelfand and its local generalizations

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In 1941, I. Gelfand proved that if a is a doubly power-bounded element of a Banach algebra A such that Sp(a) = 1, then a = 1. In [4], this result has been extended locally to a larger class of operators. In this note, we first give some quantitative local extensions of Gelfand-Hille’s results. Secondly, using the Bernstein inequality for multivariable functions, we give short and elementary proofs of two extensions of Gelfand’s theorem for m commuting bounded operators, T1,...,Tm, on a Banach space X.

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@article{Drissi1997, abstract = {In 1941, I. Gelfand proved that if a is a doubly power-bounded element of a Banach algebra A such that Sp(a) = 1, then a = 1. In [4], this result has been extended locally to a larger class of operators. In this note, we first give some quantitative local extensions of Gelfand-Hille’s results. Secondly, using the Bernstein inequality for multivariable functions, we give short and elementary proofs of two extensions of Gelfand’s theorem for m commuting bounded operators, $T_1,..., T_m$, on a Banach space X.}, author = {Drissi, Driss}, journal = {Studia Mathematica}, keywords = {locally power-bounded operator; local spectrum; local spectral radius; doubly power-bounded element of a Banach algebra; Bernstein inequality for multivariable functions; Gelfand’s theorem for commuting bounded operators}, language = {eng}, number = {2}, pages = {185-194}, title = {On a theorem of Gelfand and its local generalizations}, url = {http://eudml.org/doc/216387}, volume = {123}, year = {1997},}

TY - JOURAU - Drissi, DrissTI - On a theorem of Gelfand and its local generalizationsJO - Studia MathematicaPY - 1997VL - 123IS - 2SP - 185EP - 194AB - In 1941, I. Gelfand proved that if a is a doubly power-bounded element of a Banach algebra A such that Sp(a) = 1, then a = 1. In [4], this result has been extended locally to a larger class of operators. In this note, we first give some quantitative local extensions of Gelfand-Hille’s results. Secondly, using the Bernstein inequality for multivariable functions, we give short and elementary proofs of two extensions of Gelfand’s theorem for m commuting bounded operators, $T_1,..., T_m$, on a Banach space X.LA - engKW - locally power-bounded operator; local spectrum; local spectral radius; doubly power-bounded element of a Banach algebra; Bernstein inequality for multivariable functions; Gelfand’s theorem for commuting bounded operatorsUR - http://eudml.org/doc/216387ER -